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Theorem 2ndcsep 21262

Description: A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.)

**Hypothesis**
Ref	Expression
2ndcsep.1	⊢ 𝑋 = ∪ 𝐽

**Assertion**
Ref	Expression
2ndcsep	⊢ (𝐽 ∈ 2^nd𝜔 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))

Distinct variable groups: 𝑥,𝐽 𝑥,𝑋

Proof of Theorem 2ndcsep Dummy variables 𝑓 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		is2ndc 21249	. 2 ⊢ (𝐽 ∈ 2^nd𝜔 ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2		vex 3203	. . . . . . . . 9 ⊢ 𝑏 ∈ V
3		difss 3737	. . . . . . . . 9 ⊢ (𝑏 ∖ {∅}) ⊆ 𝑏
4		ssdomg 8001	. . . . . . . . 9 ⊢ (𝑏 ∈ V → ((𝑏 ∖ {∅}) ⊆ 𝑏 → (𝑏 ∖ {∅}) ≼ 𝑏))
5	2, 3, 4	mp2 9	. . . . . . . 8 ⊢ (𝑏 ∖ {∅}) ≼ 𝑏
6		simpr 477	. . . . . . . 8 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → 𝑏 ≼ ω)
7		domtr 8009	. . . . . . . 8 ⊢ (((𝑏 ∖ {∅}) ≼ 𝑏 ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
8	5, 6, 7	sylancr 695	. . . . . . 7 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9		eldifsn 4317	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
10		n0 3931	. . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝑦)
11		elunii 4441	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → 𝑧 ∈ ∪ 𝑏)
12		simpl 473	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → 𝑧 ∈ 𝑦)
13	11, 12	jca 554	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑏) → (𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
14	13	expcom 451	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑏 → (𝑧 ∈ 𝑦 → (𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦)))
15	14	eximdv 1846	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑏 → (∃𝑧 𝑧 ∈ 𝑦 → ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦)))
16	15	imp 445	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑏 ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
17		df-rex 2918	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑏 ∧ 𝑧 ∈ 𝑦))
18	16, 17	sylibr 224	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑏 ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
19	10, 18	sylan2b 492	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
20	9, 19	sylbi 207	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦)
21	20	rgen 2922	. . . . . . 7 ⊢ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦
22		vuniex 6954	. . . . . . . 8 ⊢ ∪ 𝑏 ∈ V
23		eleq1 2689	. . . . . . . 8 ⊢ (𝑧 = (𝑓‘𝑦) → (𝑧 ∈ 𝑦 ↔ (𝑓‘𝑦) ∈ 𝑦))
24	22, 23	axcc4dom 9263	. . . . . . 7 ⊢ (((𝑏 ∖ {∅}) ≼ ω ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 ∈ ∪ 𝑏𝑧 ∈ 𝑦) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
25	8, 21, 24	sylancl 694	. . . . . 6 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦))
26		frn 6053	. . . . . . . . 9 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ran 𝑓 ⊆ ∪ 𝑏)
27	26	ad2antrl 764	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ⊆ ∪ 𝑏)
28		vex 3203	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
29	28	rnex 7100	. . . . . . . . 9 ⊢ ran 𝑓 ∈ V
30	29	elpw 4164	. . . . . . . 8 ⊢ (ran 𝑓 ∈ 𝒫 ∪ 𝑏 ↔ ran 𝑓 ⊆ ∪ 𝑏)
31	27, 30	sylibr 224	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ∈ 𝒫 ∪ 𝑏)
32		omelon 8543	. . . . . . . . . . 11 ⊢ ω ∈ On
33	6	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34		ondomen 8860	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
35	32, 33, 34	sylancr 695	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑏 ∈ dom card)
36		ssnum 8862	. . . . . . . . . 10 ⊢ ((𝑏 ∈ dom card ∧ (𝑏 ∖ {∅}) ⊆ 𝑏) → (𝑏 ∖ {∅}) ∈ dom card)
37	35, 3, 36	sylancl 694	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ∈ dom card)
38		ffn 6045	. . . . . . . . . . 11 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → 𝑓 Fn (𝑏 ∖ {∅}))
39	38	ad2antrl 764	. . . . . . . . . 10 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓 Fn (𝑏 ∖ {∅}))
40		dffn4 6121	. . . . . . . . . 10 ⊢ (𝑓 Fn (𝑏 ∖ {∅}) ↔ 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
41	39, 40	sylib 208	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
42		fodomnum 8880	. . . . . . . . 9 ⊢ ((𝑏 ∖ {∅}) ∈ dom card → (𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓 → ran 𝑓 ≼ (𝑏 ∖ {∅})))
43	37, 41, 42	sylc 65	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ≼ (𝑏 ∖ {∅}))
44	8	adantr 481	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ≼ ω)
45		domtr 8009	. . . . . . . 8 ⊢ ((ran 𝑓 ≼ (𝑏 ∖ {∅}) ∧ (𝑏 ∖ {∅}) ≼ ω) → ran 𝑓 ≼ ω)
46	43, 44, 45	syl2anc 693	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ran 𝑓 ≼ ω)
47		tgcl 20773	. . . . . . . . . 10 ⊢ (𝑏 ∈ TopBases → (topGen‘𝑏) ∈ Top)
48	47	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (topGen‘𝑏) ∈ Top)
49		unitg 20771	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ V → ∪ (topGen‘𝑏) = ∪ 𝑏)
50	2, 49	ax-mp 5	. . . . . . . . . . 11 ⊢ ∪ (topGen‘𝑏) = ∪ 𝑏
51	50	eqcomi 2631	. . . . . . . . . 10 ⊢ ∪ 𝑏 = ∪ (topGen‘𝑏)
52	51	clsss3 20863	. . . . . . . . 9 ⊢ (((topGen‘𝑏) ∈ Top ∧ ran 𝑓 ⊆ ∪ 𝑏) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
53	48, 27, 52	syl2anc 693	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ ∪ 𝑏)
54		ne0i 3921	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑦 → 𝑦 ≠ ∅)
55	54	anim2i 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑦 ∈ 𝑏 ∧ 𝑦 ≠ ∅))
56	55, 9	sylibr 224	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓‘𝑦) ∈ ran 𝑓)
58	38, 57	sylan 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓‘𝑦) ∈ ran 𝑓)
59		inelcm 4032	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓‘𝑦) ∈ 𝑦 ∧ (𝑓‘𝑦) ∈ ran 𝑓) → (𝑦 ∩ ran 𝑓) ≠ ∅)
60	59	expcom 451	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑦) ∈ ran 𝑓 → ((𝑓‘𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
61	58, 60	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → ((𝑓‘𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
62	61	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → (𝑦 ∈ (𝑏 ∖ {∅}) → ((𝑓‘𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
63	62	a2d 29	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → (𝑦 ∈ (𝑏 ∖ {∅}) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
64	56, 63	syl7 74	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → ((𝑦 ∈ 𝑏 ∧ 𝑥 ∈ 𝑦) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
65	64	exp4a 633	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓‘𝑦) ∈ 𝑦) → (𝑦 ∈ 𝑏 → (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))))
66	65	ralimdv2 2961	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 → (∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦 → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
67	66	imp 445	. . . . . . . . . . . 12 ⊢ ((𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦) → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
68	67	ad2antlr 763	. . . . . . . . . . 11 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69		eqidd 2623	. . . . . . . . . . . 12 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → (topGen‘𝑏) = (topGen‘𝑏))
70	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ∪ 𝑏 = ∪ (topGen‘𝑏))
71		simplll 798	. . . . . . . . . . . 12 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑏 ∈ TopBases)
72	27	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → ran 𝑓 ⊆ ∪ 𝑏)
73		simpr 477	. . . . . . . . . . . 12 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑥 ∈ ∪ 𝑏)
74	69, 70, 71, 72, 73	elcls3 20887	. . . . . . . . . . 11 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → (𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓) ↔ ∀𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
75	68, 74	mpbird 247	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) ∧ 𝑥 ∈ ∪ 𝑏) → 𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓))
76	75	ex 450	. . . . . . . . 9 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → (𝑥 ∈ ∪ 𝑏 → 𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓)))
77	76	ssrdv 3609	. . . . . . . 8 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ∪ 𝑏 ⊆ ((cls‘(topGen‘𝑏))‘ran 𝑓))
78	53, 77	eqssd 3620	. . . . . . 7 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)
79		breq1 4656	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
80		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = ran 𝑓 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘(topGen‘𝑏))‘ran 𝑓))
81	80	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑥 = ran 𝑓 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏))
82	79, 81	anbi12d 747	. . . . . . . 8 ⊢ (𝑥 = ran 𝑓 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)))
83	82	rspcev 3309	. . . . . . 7 ⊢ ((ran 𝑓 ∈ 𝒫 ∪ 𝑏 ∧ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = ∪ 𝑏)) → ∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏))
84	31, 46, 78, 83	syl12anc 1324	. . . . . 6 ⊢ (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶∪ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓‘𝑦) ∈ 𝑦)) → ∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏))
85	25, 84	exlimddv 1863	. . . . 5 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏))
86		unieq 4444	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ (topGen‘𝑏) = ∪ 𝐽)
87		2ndcsep.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
88	86, 51, 87	3eqtr4g 2681	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → ∪ 𝑏 = 𝑋)
89	88	pweqd 4163	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → 𝒫 ∪ 𝑏 = 𝒫 𝑋)
90		fveq2 6191	. . . . . . . . 9 ⊢ ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
91	90	fveq1d 6193	. . . . . . . 8 ⊢ ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
92	91, 88	eqeq12d 2637	. . . . . . 7 ⊢ ((topGen‘𝑏) = 𝐽 → (((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏 ↔ ((cls‘𝐽)‘𝑥) = 𝑋))
93	92	anbi2d 740	. . . . . 6 ⊢ ((topGen‘𝑏) = 𝐽 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
94	89, 93	rexeqbidv 3153	. . . . 5 ⊢ ((topGen‘𝑏) = 𝐽 → (∃𝑥 ∈ 𝒫 ∪ 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = ∪ 𝑏) ↔ ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
95	85, 94	syl5ibcom 235	. . . 4 ⊢ ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ((topGen‘𝑏) = 𝐽 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
96	95	impr 649	. . 3 ⊢ ((𝑏 ∈ TopBases ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
97	96	rexlimiva 3028	. 2 ⊢ (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
98	1, 97	sylbi 207	1 ⊢ (𝐽 ∈ 2^nd𝜔 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ∪ cuni 4436 class class class wbr 4653 dom cdm 5114 ran crn 5115 Oncon0 5723 Fn wfn 5883 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 ωcom 7065 ≼ cdom 7953 cardccrd 8761 topGenctg 16098 Topctop 20698 TopBasesctb 20749 clsccl 20822 2^nd𝜔c2ndc 21241

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-acn 8768 df-topgen 16104 df-top 20699 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-2ndc 21243

This theorem is referenced by: met2ndc 22328

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